Notes

The "kick" of a rifle cannot be predicted. The effect is dependent on
many factors which are not readily calculated. The configuration of the
stock is very important. The drop in the stock will determine how much
muzzle rise will occur, as will how you hold the rifle. The amount of cast
off in the stock will determine (in part) whether the rifle rising in recoil
will slap you in the cheek. The construction of the butt pad will determine
how deeply the butt digs into your arm. The muzzle blast will affect your
perception of recoil. The list goes on.

Free Recoil

Estimating even the free recoil of the rifle is not simple. The free
recoil is that which would be measured if the rifle were fired suspended
from strings, free to recoil. Conservation of momentum for the portion of
the rifle's recoil due to the bullet's flight is easy enough, but to compute
ahead of time the recoil due to the propellant gases is not trivial. The
velocity at which the gases exit depends on many factors, and then there is
the matter of the surrounding air displaced by the muzzle blast.

However, there are published formulas which will compute free recoil
close enough to allow a comparison between two rifles. If one knows the
average velocity of the escaping propellant, the free recoil velocity (in a
vacuum) is simply:

V = ( b*v + c*p ) / W

where b is the bullet's weight, v the muzzle velocity,
c the charge weight, p the average velocity of the
escaping propellant gases, and W is the rifle's weight.

The NRA Fact Book (1988) gives some estimates for p. For
small arms, the gas velocity is about 4000 fps for smokeless and about 2000
for blackpowder. For cannons 4700 is used. Other references give only the
4700 fps figure.

In the British Textbook of Small Arms (1929) it is suggested one
make the propellant's velocity proportional to the bullet's:

V = ( b + k*c ) * v / W

Unfortunately, the value of k was found to vary and "lies
between 1 and 2, with an average value of 1½." If one uses that 150%,
this equation at a muzzle velocity of 2700 fps gives the same results as the
4000 fps number cited in the NRA book. It seems reasonable to me the
propellant's velocity will be related to the bullet's, but the fixed 4000
fps term gives a slightly better agreement with the recoil computations from
the internal ballistics simulator QuickLOAD.

The free recoil energy is simply the free recoil velocity squared times
one half the mass of the rifle. This energy is the figure most often given
in books and magazines.

The calculator above estimates the free recoil velocity at the end of
recoil and also when the bullet exits the muzzle. The recoil at muzzle exit
is simpler to estimate correctly. The bullet's velocity is known. The speed
of the gases varies from zero at the breech to the bullet's velocity at the
muzzle. The average velocity for the gases will, then, be roughly one half
the bullet's.

Once the bullet exits, the propellant gases rush out and cause the rifle
to recoil further, like a rocket. QuickLOAD estimates this effect is over
when a rifle in free recoil has moved about one half an inch, so this effect
will be borne by your shoulder. However, this recoil effect is smaller than
that attained at bullet exit.

The peak recoil acceleration is fairly easy to estimate but is not
useful. The maximum comes near peak chamber pressure and is computed using
that pressure times the cross sectional area of the bore, less some for
bullet friction. (The friction can be roughly estimated as a loss of 1000
psi or so from the chamber pressure.) However, this peak occurs early in the
motion of the rifle (within a few hundredths of an inch of recoil), before
the rifle has fully compressed your shoulder muscles and the butt pad, and
so you do not feel it's full effect. Consider the .308 Win. and the .300
Wea. For the same weight rifle, both have the same peak acceleration because
both have the same diameter bullet and nearly the same peak pressure, but
the .300 kicks much harder.

Recoil Sharpness

The A-Square Reloading Manual (1996) has an interesting
discussion of recoil and its effect on the body. One argument they put forth
is that the time of the recoil is neglected in the free recoil numbers.
Imagine two rifles of the same weight, both shooting cartridges of the same
momentum. If one shoots a 100 gn bullet at 3000 fps and the other a 200 gn
bullet at 1500 fps, the free recoil velocity (and energy) is similar.
However, the faster bullet will exit the barrel in much less time (roughly
half), so the acceleration (to the free recoil velocity) is much higher.
Since one's shoulder is being accelerated, and since force is mass times
acceleration, the faster cartridge will hurt more. One might say the recoil
of the faster cartridge would be "sharper."

One goal in creating this computer was to provide estimates of this
effect. Roughly speaking, the recoil sharpness might vary with the square of
the muzzle velocity, for increasing muzzle velocity increases the rifle's
recoil velocity while also decreasing the time over which the rifle recoils.
Unfortunately, using rough approximations, I was not able to find a pattern
which would fit the descriptions of relative recoil I've read. Perhaps a
later version of this calculator will show some progress.

Barsness' Index

In 2007, the writer John Barsness posted on an internet forum the equations behind a
recoil estimator he devised about 10 years earlier. Given both his wide
experience in the shooting sports and the quality of his writings, his
proposal should be considered.

His recoil factor starts with the kinetic energy of the bullet. This is
then scaled by the ratio of the combined weights of the bullet and charge to
that of the rifle. Lastly, a "rocket factor" is applied; this factor takes
into account the case size relative to the area
of the bore.

The physics behind the "rocket factor" was not offered, but one can see
it's utility. Since typical rifle barrel lengths don't vary greatly, the
relative capacity can serve as a stand in for the expansion ratio, and so it
provides some estimate of the muzzle pressure. It might also indicate the
width of the peak pressure pulse. By starting with the bullet's kinetic
energy, Barsness' formula incorporates the square of the bullet velocity,
and as mentioned in the previous section, this might be a factor in recoil
sharpness.

Inputs and Outputs

To more easily judge how the numbers come out for typical sporters, a
number of sample inputs are available. The numbers for the .308 and the .300
were selected to match the inputs Barsness gave in his examples. The other
samples are based on numbers from the QuickLOAD simulator and from the Hodgdon 2004 Annual.

Among the sample inputs, the rifle's weight varies some, with heavier
weights assumed for large bore rifles. If the box next to the input for
rifle weight is checked, this weight will not be altered by selecting one of
the sample inputs.

For the average speed of the ejected propellant gases, the calculator
uses 1.5 times the bullet's fps by default, but the other estimates for gas
velocity can be selected instead. A fixed velocity can be selected, or the
velocity can be made proportional to that of the bullet (ie. use the
factor k suggested by the Textbook of Small Arms).

The momentum, velocity, and kinetic energy are estimated for the rifle
at the end of recoil. The velocity and kinetic energy of the rifle are also
estimated at the point the bullet is exiting the barrel. The rocket effect
accounts for the difference in the two sets of numbers.

Barsness' recoil index is calculated when the cartridge's bullet
diameter and net case capacity (in gn of water) are input with the other
numbers. The Powley Computer on this site can be
used to calculate the net case capacity of a cartridge, or you can just
knock about 10% off the capacity of the empty case.

Note: JavaScript must be enabled for the calculator to work.

Conclusions

I've not had the opportunity to shoot a wide range of rifles, so I do
not have a feel for how well free recoil numbers predict perceived recoil.
I'm skeptical, for a rifle on your shoulder is clearly not in free recoil.

The recoil momentum does tell you how much the gun will shove you
around. I suspect free recoil velocity might be a useful indicator of recoil
harshness, and others do as well. In the 08/2003 issue of Handloader
magazine, Steve Gash cites a 1931 work on shotguns which mentioned a recoil
of about 16 fps is "too great to be withstood." This was probably for
shoulder guns with steel butt pads, for many factory rifles offered today
exceed 16 fps of free recoil. A .375 recoils at about 16 fps, and most
shooters can learn to tolerate this level of recoil.

On the other hand, others from that era noted velocity alone is not the
figure of merit. From the Textbook of Small Arms:

As regards the sensation of recoil, it seems well established
that the actual velocity of recoil of the gun is a very great factor. In
shot guns weighing 6 to 7 lbs, 15 fps has been long established as a maximum
above which gun-headache is sure to ensue. But with an elephant rifle
weighing perhaps 15 lbs, such a velocity is unbearable for more than one or
two shots.

For reference, the shotguns mentioned would have a free recoil momentum
of about 100 lb-fps and an energy of about 25 ft-lb, and the elephant rifle
would give numbers somewhat over twice these.

In response to Barsness' post about his recoil index, several shooters
responded they've found his index to be a more reliable indicator than other
formulas. Barsness' index is closer to the numbers for free recoil energy,
so my hunch that recoil velocity was a good figure of merit is not
supported. Although the approach is different, computing free recoil energy
by setting the velocity of the gases to either 1.50 or 1.75 times that of
the bullet produces numbers which track Barsness' index.